1. Field of the Invention
The invention relates to a method for calculation of parameters for frequency-modulated and/or amplitude-modulated electromagnetic pulses for adiabatic transfer processes in a quantum-mechanical system, with the dynamic processes in the quantum-mechanical system being described by a Hamilton operator H(t) in an instantaneous coordinate system.
2. Background Description
The magnetism of the material is based on magnetic moments {right arrow over (m)} of elementary particles of which the material is composed, in which the magnetic moments can interact with an external magnetic field. When the external magnetic field is static, the magnetic moments carry out a precession movement. The precession frequency of the magnetic moments depends on the magnetic field strength. If a magnetic alternating field is now also applied in addition to the static magnetic field, at a frequency which is matched to the precession frequency of the magnetic moments, a resonant absorption of energy from the magnetic alternating field occurs. All phenomena which are subsumed by the expression “magnetic resonance” are based on this principle.
The reason for the magnetic moments of the elementary particles is their total angular momentum, which includes the spin as a quantum-mechanical variable. The theoretical treatment of magnetic resonance is thus based on quantum mechanics. The dynamic response of a quantum-mechanical system is described by what is referred to as the Hamilton operator H, which is included in what is referred to as the Schrödinger equation. The Schrödinger equation is an equation of motion for the state vector which describes the state of a quantum-mechanical system. For a definition of the Schrödinger equation and of the Hamilton operator, reference shall be made to the Lexikon der Physik [Dictionary of Physics], Volumes 3 and 5, Spektrum Akademischer Verlag, Heidelberg, 2000.
Frequency-modulated and amplitude-modulated electromagnetic pulses are used for adiabatic polarization and coherence transfer in atomic and molecular spectroscopy (laser optics, magnetic resonance) and in imaging processes (magnetic resonance tomography, magnetic resonance imaging). One problem which occurs in trials is the random interaction between the elementary particles, and this leads to relaxation processes which force the system back to the equilibrium state after a deflection. Secondly, theoretically fixed values are assumed for the system variables, such as the transition frequency, field amplitude etc. even though these variables may be subject to fluctuations, or their values may not be sufficiently well known, in a macroscopic sample.
Adiabatic techniques are therefore used, and are particularly advantageous when a spectroscopic parameter has a broad bandwidth (variation in the intensity of the molecular coupling, distribution of transition frequencies produced by field gradients), or when the aim is to avoid a difficult adjustment (homogeneity of external fields).
The pulse shapes and parameters of the frequency-modulated and/or amplitude-modulated electromagnetic pulses are conventionally determined empirically. A successful pulse shape, satisfies what is referred to as the adiabatic condition, which is required as a control criterion in this case. This adiabatic condition is a criterion in the form of an inequality for what is referred to as the adiabatic approximation, and can be formulated in various ways. If, for example, the square of the magnitude of the quotient of the maximum angular velocity of an eigen state (energy level) and of the minimum transition frequency to an adjacent eigen state (energy level) is very much smaller than unity
      (                                                                α              max                                      ω              min                                                2            ⁢              <<        1              )    ,the adiabatic approximation is regarded as being valid. The expression adiabatic, that is to say the dynamic response in the sense of the adiabatic approximation, is defined by Ehrenfest such that the change in the system during the action of the electromagnetic field is so slow that the change in the energy level is negligibly small. In other words, in the situation under consideration of the magnetic moments, the magnetization follows the magnetic field, provided that the field direction changes sufficiently slowly.
The sudden and adiabatic change in the Hamilton operator is explained theoretically in detail in Albert Messiah, Quantenmechanik [Quantum mechanics], Volume 2, 2nd edition, Walter De Gruyter, Berlin, N.Y. 1985, Chapter 17.2, pages 223 to 236.
The adiabatic approximation corresponds to a simplification in the description of the dynamic response of the system, by ignoring any coupling between a radiation field and the system that is caused by variation with time of the carrier frequency, and of the amplitude. The dynamic response is then approximately stationary, that is to say it is assumed that the carrier frequency and the amplitude of the input radiation do not vary with time, but that their instantaneous values are constant. The adiabatic condition can always be satisfied for a given pulse shape if the rate of change of the modulation functions is made sufficiently small. However, this correspondingly lengthens the duration of the desired polarization or coherence transfer. B. Shore, The Theory of Coherent Atomic Excitation, John Wiley, New York 1990, page 303, discloses the determination of pulse shapes from the need to obtain analytical solutions to the equation of motion, with the adiabatic dynamic response being achieved only in a second step by slowing down the modulation processes and hence the desired transfer.
In other approaches, the adiabatic condition is transformed from the form of an inequality, which is not suitable for calculation of modulation functions, to the form of an equation with the assistance of further physical assumptions. This can be regarded as a conditional equation in the sought functions and allows them to be determined, possibly with the assistance of secondary conditions. Once again, the rate of change of the modulation functions must be determined empirically in order to lie in the validity range of the adiabatic approximation. These approaches are described, for example, in                C. J. Hardy, W. A. Edelstein and D. Vatis, Journal of Magnetic Resonance 66, 470, 1986;        A. Tannus and M. Garwood, Journal of Magnetic Resonance A 120, 133, 1996; and        D. Rosenfeld and Y. Zur, Magnetic Resonance in Medicine 36, 124, 1996.        
Pulse shapes for the two-level system (spin ½) are typically developed, in which there are no internal interactions and the dynamic response is governed exclusively by the applied radiation, and hence can be controlled completely.
The conventional determination of pulse shapes for adiabatic polarization and coherence transfers have the following disadvantages:
The semi-empirical methods do not take account of the relaxation processes to which the system is subject. A successful adiabatic transfer has to take place in a shorter time than that in which the system relaxes significantly. It is thus desirable to use fast modulations. However, this contradicts the fundamental adiabatic approximation. Optimum pulse shapes for the fastest possible transfer must therefore be found in the context of the adiabatic condition. However, the adiabatic condition, as an inequality, defines only a relatively broad frame for this object. It is thus difficult to determine pulse shapes which carry out a desired adiabatic transfer virtually optimally, that is to say in a very short time over a given parameter bandwidth. Conventionally, attempts have thus been made to optimize the parameters for frequency-modulated and/or amplitude-modulated electromagnetic pulses for adiabatic transfer processes by experiments. This complex procedure need not necessarily, be successful which leads to a comparatively reduced signal-to-noise ratio. In the case of systems which relax sufficiently quickly, adiabatic techniques may not be used at the moment.
In addition to the relaxation-dependent losses, what are referred to as diabatic losses occur as a consequence of an imperfect adiabatic transfer. These are caused by the coupling that has been ignored, since the actual dynamic response of the system is not the same as the theoretical dynamic response as assumed for the adiabatic approximation. The magnitude of the diabatic losses is governed by the quality of the adiabatic condition. The diabatic losses increase as the transfer processes become faster.
For inherently time-dependent systems in which time-dependent interactions occur even without any input radiation, it is very difficult to find any suitable pulse shapes whatsoever for induction of an adiabatic dynamic response.
In addition, there are no pulse shapes which have been developed especially for adiabatic polarization and coherence transfers in multi-level systems, possibly with internal interactions. Processes such as these have until now been carried out using modulation functions which were determined for a two-level system.
WO 99/355 09 describes a method for calculation of parameters for frequency-modulated and/or amplitude-modulated pulses. The principle is to find a matching velocity profile for a given trajectory, such that the corresponding pulse induces an adiabatic dynamic response in a coordinate system other than a rotating coordinate system. In this case, known trajectories are copied from a rotating coordinate system to the relevant different coordinate system, and the velocity profiles are optimized numerically.
The two variables sought for definition of a pulse, namely the frequency modulation and amplitude modulation as a function of time, can, finally, be calculated uniquely from the variables determined in this way, from the trajectory and from the velocity profile. However, this has the disadvantage that new pulse shapes must be assumed and investigated empirically. Furthermore, the trajectory and velocity profile reflect local (infinitesimal) characteristics of the resultant pulse. Both variables are also used in order to satisfy an adiabatic condition in the sense of the adiabatic approximation—that is to say ignoring interactions—in a coordinate system other than a rotating coordinate system.